Tensors admitting an expression as the sum of at most s rank one tensors can be considered points of the sth order secant variety of a Segre variety, yet the most basic invariant of this variety—its dimension—is not yet fully understood. A conjecture was nevertheless proposed by Abo et al. (2009, Trans. Amer. Math. Soc., 361, 767–792), proved to be correct for s < 7. We propose a numerical randomized algorithm for testing whether a mathematically exact and structured matrix is singular requiring only the availability of an approximate matrix-vector product. Using this method we test whether the aforementioned conjecture is true. The proposed method requires several orders of magnitude less memory than approaches based on symbolic arithmetic, thus greatly increasing the number of varieties that can be handled. Our experiments establish that the Segre varieties PC^{n_1} × PC^{n_2} × ··· × PC^{n_d} embedded in PC^{n_1 × ··· × n_d} with expected generic rank s < 56 obey the conjecture; the probability that a defective variety is incorrectly classified as nondefective is less than 10^{−55}.